The number of distinct elements or members in a finite set is known as the cardinal number of a set. Basically, through cardinality, we define the size of a set. The cardinal number of a set A is denoted as n(A), where A is any set and n(A) is the number of members in set A.
Consider a set A consisting of the prime numbers less than 10.
Set A ={2, 3, 5, 7}.
As the set A consists of 4 elements, therefore, the cardinal number of set A is given as n(A) = 4.
Properties related to difference, union and intersection and the cardinal number of set
i) Union of Disjoint Sets:
If A and B are two finite sets and if A ∩ B = ∅, then
n(A ∪ B) = n(A) + n(B)
In simple words if A and B are finite sets and these sets are disjoint then the cardinal number of Union of sets A and B is equal to the sum of the cardinal number of set A and set B.
Figure 1- Disjoint sets
The union of the disjoint sets A and B represented by the Venn diagram is given by A ∪ B and it can be seen that A ∩ B = ∅ because no element is common to both the sets.
ii) Union of two sets:
If A and B are two finite sets, then
n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
Simply, the number of elements in the union of set A and B is equal to the sum of cardinal numbers of the sets A and B, minus that of their intersection.
Figure 2- Union of two sets
In the figure given above the differently shaded regions depict the different disjoint sets i.e. A – B, B – A and A ∩ B are three disjoint sets as shown and the sum of these represents A ∪ B. Hence,
n (A ∪ B) = n (A – B) + n(B – A) + n(A ∩ B)
iii) Union of three sets
If A, B and C are three finite sets, then;
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C)
This is clearly visible from the Venn diagram that the union of the three sets will be the sum of the cardinal number of set A, set B, set C and the common elements of the three sets excluding the common elements of sets taken in pairs of two.
Example: There is a total of 200 students in class XI. 120 of them study mathematics, 50 students study commerce and 30students study both mathematics and commerce. Find the number of students who
i) Study mathematics but not commerce
ii) Study commerce but not mathematics
iii) Study mathematics or commerce
Solution: The total number of students represents the cardinal number of the universal set. Let A denote the set of students studying mathematics and set B represent the students studying commerce.
Therefore,
n (U) = 200
n(A) = 120
n(B) = 50
n(A ∩ B) = 30
The Venn diagram represents the number of students studying mathematics and commerce. i) Here, we are required to find the difference of sets A and B.
n(A) = n(A – B) + n(A ∩ B)
n(A-B) = n(A) –n(A ∩ B) ⇒ n (A – B) = 120 – 30 = 90
The number of students who study mathematics but not commerce is 90.
ii) Similarly here, we are required to find the difference of sets B and A
n (B) = n (B – A) + n (A ∩ B) ⇒ n (B – A) = 50 – 30 = 20
The number of students who study commerce but not mathematics is 20.
iii) The number of students who study mathematics or commerce
n (A ∪ B) = n(A) + n(B) – n(A ∩ B)
⇒ n(A ∪ B) = 120 + 50 – 30 = 140
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The union function of two sets has all the elements or objects present in two sets or either of the two sets. It is represented by ⋃. The intersection function of two sets is when all the elements present in the both sets are present. It is represented as ⋂.
Consider two sets A and B. A = {2, 4, 5, 6,10,11,14, 21}, B = {1, 2, 3, 5, 7, 8,11,12,13} and A ∩ B = {2, 5, 11}, and the cardinal number of A intersection B is represented by n(A ∩ B)= 3. n(A ∩ B)= n(A) + n(B) - n(A ∪ B)
n(A ∪ B)= n(A) + n(B) – n(A ∩ B) Simply, the number of elements in the union of set A and B is equal to the sum of cardinal numbers of the sets A and B, minus that of their intersection.
The intersection of two sets A and B, denoted A∩B, is the set of elements common to both A and B. In symbols, ∀x∈U[x∈A∩B⇔(x∈A∧x∈B)]. The union of two sets A and B, denoted A∪B, is the set that combines all the elements in A and B. In symbols, ∀x∈U[x∈A∩B⇔(x∈A∨x∈B)].
What are the various cardinal properties of sets? Ans: The various cardinal properties of n ( A ∪ B ) = n ( A ) + n ( B ) − n ( A ∩ B ) , n ( A − B ) = n ( A ) − n ( A ∩ B ) and if A ∩ B = ϕ , then n ( A ∪ B ) = n ( A ) + n ( B ) .
First, we count the total number of elements in the set.We then group all the same elements in one group.Number of such groups made is called cardinal number of a set.
What are examples of union and intersection? For a union and an intersection example, use the set B = {1, 2, 3, 4, 5, 6} and the set D = {3, 5, 7, 9, 10}. The union of sets B and D is the set {1, 2, 3, 4, 5, 6, 7, 9, 10}. The intersection of sets B and D is the set {3, 5}.
To find the intersection of two or more sets, you look for elements that are contained in all of the sets. To find the union of two or more sets, you combine all the elements from each set together, making sure to remove any duplicates.
The cardinal numbers are the counting numbers that start from 1 and go on sequentially and are not fractions. The examples of cardinal numbers are: 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,…. The meaning of cardinals is “how many” of anything is existing in a group.
Step 1: Determine all of the elements in the first set. Step 2: Determine all of the elements in the second set. Step 3: The intersection is formed by including all of the elements that appear in both Step 1 and Step 2.
To calculate the cardinality of a set, count the number of distinct elements in the set. For example, if you have a set A = {1, 2, 3, 3, 4}, the cardinality of set A is 4, as there are four distinct elements.
The cardinality of a set is nothing but the number of elements in it. For example, the set A = {2, 4, 6, 8} has 4 elements and its cardinality is 4. Thus, the cardinality of a finite set is a natural number always.
The union of two sets is a new set that contains all of the elements that are in at least one of the two sets. The union is written as A∪B or “A or B”. The intersection of two sets is a new set that contains all of the elements that are in both sets. The intersection is written as A∩B or “A and B”.
Given four sets A, B, C and D, the formula for the union of these sets is as follows: P (A U B U C U D) = P(A) + P(B) + P(C) +P(D) - P(A ∩ B) - P(A ∩ C) - P(A ∩ D)- P(B ∩ C) - P(B ∩ D) - P(C ∩ D) + P(A ∩ B ∩ C) + P(A ∩ B ∩ D) + P(A ∩ C ∩ D) + P(B ∩ C ∩ D) - P(A ∩ B ∩ C ∩ D).
The not-element-of symbol looks like the element-of symbol except that a forward slash runs through it (∉). The not-element-of symbol is read as "is not an element of," "is not a member of," "is not in" or "does not belong to." For example, the following expression indicates that 7 is not an element of set A: 7 ∉ A.
The intersection of sets contains elements common to all sets, while the union includes all unique elements from all sets. For three sets A, B, and C, the intersection (A B C) contains elements present in all three sets, whereas the union (A B C) includes elements from any of the sets.
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